Here are two designs on the same theme, based on an idea originally due to Geoffrey Irving (who is involved in 3D printing, but isn't on Shapeways as far as I know). The idea is to show how the fractal curve is generated, by illustrating the iterative process as the curve gets more and more crinkly as a surface.

As I said in the original post, I also stole the idea from Geoffrey Irving

We wrote a paper together on these things: Developing Fractal Curves. Geoffrey has a different way to generate the surfaces, which looks like it might be similar to yours. He also did the Koch snowflake, there are some renders of it in the paper.

Thank you for the paper.
We both use polynomial curves but I don't know if it is the same algorithm..
And you manage to create thickness (I have to use Blender) AND it gets thinner as the iteration (or z space parameter) increase.
Doing that would be my next step .

The details about how we thicken things up are in the paper, but there are lots of ways you can do it.

I made my version with NURBS surfaces, so it can be however many polygons you like. I tend to get near to Shapeways' million polygon limit if I've got a complicated model like this. Geoffrey's version of these uses meshes the whole time, and gets finer detail meshes using something called Loop subdivision, and so he can get arbitrarily detailed meshes as well.